Problems and Solutions in Mathematical Finance: Stochastic Calculus, Volume I

2.2.4 First Passage Time

1. 1. Doob's Maximal Inequality. Let be a probability space and let be a continuous non-negative submartingale with respect to the filtration , . Given and , show that
   

   By writing

   

   show that

   

   Deduce that if is a continuous non-negative supermartingale with respect to the filtration , then

   Solution

   For we let so that . Because is a non-negative submartingale we have ...